WARM UP Find the angle measurement: 1. m JKL 127° L x° K  J m JKL = 127

PROVING LINES PARALLEL

OBJECTIVES Use the angles formed by a transversal to prove two lines are parallel

DEFINITION The converse of a theorem is the opposite of that theorem. An example of a converse theorem:

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. 4 ≅ 8 4 ≅ 8 4 and 8 are corresponding angles. ℓ || m Converse of Corresponding Angles Postulate EXAMPLE 1

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m 3 = (4x – 80)°, m 7 = (3x – 50)°, x = 30 m 3 = 4(30) – 80 = 40 Substitute 30 for x. m 8 = 3(30) – 50 = 40 Substitute 30 for x. ℓ || m Conv. of Corr.  s Post. 3 ≅ 8 Def. of ≅ angles. m 3 = m 8 Trans. Prop. of Equality EXAMPLE 1

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m 1 = m 3 1 ≅ 3  1 and  3 are corresponding angles. ℓ || m Conv. of Corr. Angles Post. TRY THIS…

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m 7 = (4x + 25)°, m 5 = (5x + 12)°, x = 13 m 7 = 4(13) + 25 = 77 Substitute 13 for x. m 5 = 5(13) + 12 = 77 Substitute 13 for x. ℓ || m Conv. of Corr. Angles Post. 7 ≅ 5 Def. of ≅ angles m 7 = m 5 Trans. Prop. of Equality EXAMPLE 2

The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. PARALLEL POSTULATE

MORE THEOREMS

Use the given information and the theorems you have learned to show that r || s. 4 ≅ 8 4 ≅ 8 4 and 8 are alternate exterior angles. r || s Conv. Of Alt. Int. Angles Theorem PARALLEL LINES

m 2 = (10x + 8)°, m 3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. m 2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x. m 3 = 25x – 3 = 25(5) – 3 = 122 Substitute 5 for x. EXAMPLE 3

m 2 = (10x + 8)°, m 3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. r || s Conv. of Same-Side Int. Angles Theorem. m 2 + m 3 = 58° + 122° = 180° 2 and 3 are same-side interior angles. EXAMPLE CONTINUED

m 4 = m 8 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. 4 ≅ 8 4 and 8 are alternate exterior angles. r || s Conv. of Alt. Int. Angles Theorem. 4 ≅ 8 Congruent angles TRY THIS…

Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m 3 = 2x°, m 7 = (x + 50)°, x = 50 m 3 = 100° and m 7 = 100° 3 ≅ 7 r||s Conv. of the Alt. Int. Angles Theorem. m 3 = 2x = 2(50) = 100° Substitute 50 for x. m 7 = x + 50 = = 100° Substitute 5 for x. EXAMPLE 4

A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m 1= (8x + 20)° and m 2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel. CARPENTRY APPLICATION

A line through the center of the horizontal piece forms a transversal to pieces A and B. 1 and 2 are same-side interior angles. If 1 and 2 are supplementary, then pieces A and B are parallel. Substitute 15 for x in each expression. SOLUTION

m 1 = 8x + 20 = 8(15) + 20 = 140 m 2 = 2x + 10 = 2(15) + 10 = 40 m 1+m 2 = = 180 Substitute 15 for x. 1 and 2 are supplementary. The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem. SOLUTION CONT.

Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel. 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30° The angles are congruent, so the oars are || by the Converse of the Corresponding Angles Postulate. TRY THIS…

HOMEWORK Textbook pg. 165 “Think & Discuss” Textbook pg. 166 #2, 6, 8, 14,18 & 24